I'm self-employed, in the US. I run websites professionally. I get my
income from advertisements essentially. I make quite a bit more than I
spend, and I want to invest my money. since I know the niche so well,
I figured a great place to re-invest my money would be in other
websites. I've heard other internet professionals do similar things.

however, when I talked to my accountant, he said I can't write off the
purchase price of existing websites as a business expense because it's
not a tangible object. he mentioned something about writing off
expenses from these types of purchases over a series of years, but
that didn't sound terribly attractive.

is there any way at all for me to buy websites to expand my business
and have the purchases be exempt from federal tax?

You've got problems with the terms you're using.  When you spend money
on an asset that you invest in for your business, it's not an expense.
 It's an expenditure but not an expense for the current tax year.

Also, when you say "purchases be exempt from federal tax", that does
not compute.  Purchases aren't exempt or non-exempt from taxation. 
Income is taxed not purchases or sales (per se) or investments. 
Income (Revenues minus expenses basically) is taxed, capital gains
(proceeds from sale minus cost of purchase basically) is also taxed.

Also, when you say you want to "invest your money", this is not really
what you're meaning to say, I don't believe.  The reason is that you
can't expense investments.  Therefore, "investing" your money wouldn't
lower your taxable income.  (other than through depreciation as
discussed below).

So, I believe that if you have a more fundamental understanding of
basic accounting, you'll start answering your own questions.  Since
I'm not a researcher I won't spend a lot of time going into detail but
here are a few thoughts to pursue.

In my first paragraph, I mention buying an asset.  If you buy an asset
for use in your business such as a piece of equipment, it should be
capitalized and depreciated.  These are items that are used up over
more than 1 year as opposed to supplies or utilities costs that would
be used up current and would therefore be expensed.

Capitalizing an item basically means that it's put on the balance
sheet and depreciated over time.  If you buy a printer for $1000 and
it has a 5 year life (these are identified in IRS publication #946)
then you would (depending on the depreciation method) depreciate (ie
expense) $200 per year.  The depreciation expense is the allocation of
the equipment's cost to the current year.  Depreciating is a way of
spreading the cost of the item over it's useful life rather than all
at once which would not make sense if you ponder it.

Those are some quick thoughts.  I'm sure one of the researchers will
have some additional and more specific thoughts.